De-embedding method for a sensing area characterization of planar microstrip sensors without evaluating error networks

A de-embedding method for determining all scattering (S-) parameters (e.g., characterization) of a sensing area of planar microstrip sensors (two-port network or line) is proposed using measurements of S-parameters with no calibration. The method requires only (partially known) non-reflecting line and reflecting line standards to accomplish such a characterization. It utilizes uncalibrated S-parameter measurements of a reflecting line, direct and reversed configurations of a non-reflecting line, and direct and reversed configurations of the sensing area. As different from previous similar studies, it performs such a characterization without any sign ambiguity. The method is first validated by extracting the S-parameters of a bianisotropic metamaterial slab, as for a two-port network (line), constructed by split-ring-resonators (SRRs) from waveguide measurements. Then, it is applied for determining the S-parameters of a sensing area of a microstrip sensor involving double SRRs next to a microstrip line. The root-mean-square-error (RMSE) analysis was utilized to analyze the accuracy of our method in comparison with other techniques in the literature. It has been observed from such an analysis that our proposed de-embedding technique has the lowest RMSE values for the extracted S-parameters of the sensing area of the designed sensor in comparison with those of the compared other de-embedding techniques in the literature, and have similar RMSE values in reference to those of the thru-reflect-line calibration technique. For example, while RMSE values of real and imaginary parts of the forward reflection S-parameter of this sensing area are, respectively, around 0.0271 and 0.0279 for our de-embedding method, those of one of the compared de-embedding techniques approach as high as 0.0318 and 0.0324.


The analysis of the method
Five different measurement configurations composing of two different standards and one device in the implementation of our method are schematically depicted in Fig. 2. Figure 2a corresponds to the configuration where a (reciprocal) reflecting line (R-Line) is connected between two unknown error networks X and Y, which are complex functions of VNA source and load mismatches, impedance change at the connections, SMA connectors and tapers or launchers, feed lines, etc. Figure 2b illustrates the configuration where a non-reflecting Figure 1.A planar microstrip sensor with a sensing area (double ring resonators) with feedlines and SMA tapers or launchers.Here, T X and T Y account for the effects of microstrip lines, SMA tapers or lauchers, coaxial lines with SMA connectors, and VNA systematic errors.
Table 1.Comparison of the proposed method ('PM') with other calibration and de-embedding techniques in the literature.

Calibration Technique De-embedding Technique
Parameter SOLT 21 TRL 22 (or LRL) 46 47-49  www.nature.com/scientificreports/line (NR-Line) is positioned next to the R-Line between X and Y. Figure 2c,d present the direct and reversed configurations of the device between X and Y. Finally, Fig. 2e demonstrates the reversed configuration in Fig. 2b.

Wave-cascaded matrix representation
The well-known wave-cascaded matrix (WCM) representation based on 8-term error model (directivity, source match, reflecting tracking, transmission) could be used to examine the theoretical analysis 22 .For the configurations in Fig. 2a-e, one obtains where M a , M b , M c , M d , and M e are the WCMs corresponding to the configurations in Fig. 2a-e, respectively; T X and T Y are the WCMs of the error networks X and Y; T RL and T NRL are the WCMs of the reflecting-and non- reflecting lines; and T D and T inv D are the WCMs of the direct and reversed configurations of the device.M a , M b , M c , M d , and M e are related to measured S-parameters: where k is a, b, c, d, or e.Using (3), it is possible to express T RL , T NRL , T D , and T inv D as Here, P and Ŵ are the propagation factor of and the first reflection coefficient at the R-Line; P 0 is the propagation factor of the NR-Line; z eff , γ eff , ε eff , and L r are the effective normalized impedance, effective propagation constant, effective permittivity, and length of the R-Line while γ eff,0 , ε eff,0 , and L nr are the effective propagation constant, effective permittivity, and length of the NR-Line; and S D 11 , S D 21 , S D 12 , and S D 22 are the S-parameters of the device.

Elimination of the effects of error matrices X and Y
Using WCMs in (1) and ( 2), it is possible to eliminate T Y 28,30 : where ' ⋆ −1 ' denotes the inverse of the square matrix ' ⋆' .
Using the trace operation of a square matrix, which corresponds to the sum of its eigenvalues, one can eliminate the effect of T X and determine 28,30 (1) (9) Vol.:(0123456789) www.nature.com/scientificreports/

Obtaining information about calibration standards
We determined from ( 4) and (16)   Correct sign in ( 23) can be specified after evaluating γ eff,0 and enforcing ℜe{γ eff,0 } ≥ 0 for a passive non-reflecting line.In fact, it is possible to obtain from ( 4) and (17)   Taking into account that the reflecting line is reciprocal (that is, 1 3 + 2 2 = 1 ), then one can derive This means that considering (3) and ( 4), only one information about the S-parameters (e.g., S 11 or S 21 ) of a reflecting reciprocal line is needed in the implementation of our method.
One can find D from ( 31)-( 33) that ( 16) (31) Here, 1 and 2 are the quantities in functions of Ŵ and P of the reflecting line in ( 4) and ( 5).
After, S D 11 is uniquely found from ( 35) Once S D 11 is calculated from (40), S D 22 and D can be determined in a simple manner from ( 34) and (35).Finally, S D 21 and S D 12 can be evaluated from ( 26)- (30).It is noted that ( 23) can be utilized to determine γ eff,0 as a byproduct.Finally, it should be stressed that when P 0 approaches unity, as other calibration methods such as the TRL calibration technique 22 , our proposed method breaks down ( 2 = 4 = 5 and 3 = 6 ).Therefore, it is not possible to determine meaningful S D 11 , S D 21 , S D 12 , and S D 22 from ( 26)- (40).Discussion of this point is given in Section Validation.

Validation Measurement setup
The rectangular waveguide setup operated at X-band (8.2-12.4GHz, a = 22.86 mm, b = 10.16 mm, and f c = 6.557GHz) was constructed for validation (see Fig. 3c).The VNA used in our measurements (Keysight Technologies -N9918A) has a frequency range between 30 kHz and 26.5 GHz.Two longer phase stable coaxial lines were employed to carry signals.Besides, two coax-to-waveguide adapters were secured to two longer additional waveguide straights (approximately 200 mm) to suppress high-order modes, if present.Details about the measurement setup are available in 51 .

Constructed bianisotropic metamaterial (MM) Slab
S-parameters of a bianisotropic metamaterial (MM) slab loaded into an X-band rectangular waveguide section, considered as for the device, were performed for the validation of our method.This slab was constructed by a unit cell with a square edge-coupled split-ring-resonator (SRR), as shown in Fig. 3a with the following geometrical parameters: L m = 2.00 mm, w = g = 0.30 mm, u x = d + t m , u y = 2.54 mm, and u z = L sub = 8.10 mm.Here, t m = 35µ m corresponds to the metal thickness (copper material with conductivity σ = 5.8 × 10 7 S/m) while d = 1.50 mm and L sub denote, respectively, the substrate thickness and length (FR4 material with ε r,sub = 4.3(1 − i0.025) ).Each sub-unit (four SRRs positioned in the y−direction) was fabricated using the conventional printed circuit technology 51 .As shown in Fig. 3b, the MM slab was formed by locating fourteen sub-units in cascaded manner in the x−direction.This slab is identical to that in the study 52 , which is just used here for validation.
The reason for using four SRRs in the y−direction, with fourteen times repetition in the x−direction, was to ensure a homogeneous material.According to the effective medium theory 51 , repetition periodicity of the unit cell over a transverse plane should be smaller than one-tenth of the wavelength in order for the intrinsically inhomogeneous MM slab to be considered as a homogeneous MM slab.In our case, the constructed MM slab has u x = 1.535 mm and u y = 2.54 mm where u x and u y are the periodicities in the x− and y−directions.Both u x and u y are considerably less than the wavelength of free-space (around 30 mm) at the middle frequency.This means that the MM slab satisfies the effective medium assumption.Two extra FR4 substrates (without any metallic design) with 10.16 × 8.10 × 0.55 mm 3 were inserted at the left and right side guide walls for eliminating air gap effect 51,52 .
While electromagnetic wave is propagating through the MM slab inside a rectangular waveguide, it interacts with the edge-coupled SRRs.This interaction interaction occurs in the following manner 53 .Electric field of the ( 34) dominant TE 10 mode in the y−axis (normal to the slit axis) forces charges with opposite polarities to accumulate at opposite sides (w.r.t. the z−axis) of both rings (electric excitation).This will in turn produce circulating currents and then create a magnetic dipole in the x−axis.Figure 3e illustrates electric field distribution (at the instant of maximum variation) on the plane of SRRs (electric flux lines originating from and ending with the SRRs) at the frequency of 11.867 GHz around which transmission S-parameter ( S 21 ) has a dip 51 .Besides, magnetic field of the dominant TE 10 mode in the x−direction, which is normal to the plane of SRRs, influences charges to circulate within the metal of the SRRs (magnetic excitation).This in turn will induce a non-zero net electric dipole moment in the y−axis.Figure 3f presents surface current distribution (at the instant of maximum variation) on the surface of the metals of SRRs (circulating current) at the same frequency (11.867GHz).As a consequence of such coupling mechanism of electric and magnetic fields over the waveguide cross section, a non-zero magneto-electric coupling will be present 53 , resulting in a non-identical forward and backward reflection S-parameters 51 .www.nature.com/scientificreports/with dotted lines) of the constructed bianisotropic MM slab.In application of the TRL calibration technique 22 , a waveguide section with a length of 9.4 mm was utilized as for the line standard.Then, calibrated S-parameters of the constructed bianisotropic MM slab were measured.In application of our method, we implemented the measurement configurations in Fig. 2a-e using uncalibrated S-parameter measurements with (and without) the rolling average (RA) procedure applied for frequency range of approximately 42 MHz 54 calculated from N int (f max − f min )/N f where N int , f max , f min , and N f denote, respectively, the number of intervals (the number of frequency points), maximum and minimum frequencies the measurements are conducted, and the number of total intervals (frequency points).In measurements, N int = 10 (deliberately selected partly greater than the value used in the study 54 for better smoothed data), f max = 12.4 GHz, f min = 8.2 GHz, and N f = 1001.While an empty waveguide section with a length of L nr = 9.4 mm was used as for the NR-Line, a waveguide section with a length of L r = 7.7 mm with a polyethylene (PE) sample (3.85 mm) flushed at its right terminal was considered as the R-Line.In selection of the length of the NR-Line, as discussed in Section The Analysis of the Method, we considered the point that P 0 does not approach unity.In obtaining simulated S-parameters, the Computer Simulation Technology (CST) Microwave Studio was utilized 51 .E t = 0 boundary conditions were applied over the transverse plane ( x = 0 , x = a , y = 0 , and y = b planes) to imitate hollow metallic waveguide.), which has S D 11 = S D 22 due to bianisotropic behavior 51 , are in good agreement with each other over entire frequency band.This validates our proposed method.Relatively smaller discrepancies between the simulated and measured/ extracted S-parameters are chiefly a cause of fabrication process 51 .Because our method assumes that P 0 = 1.0 , it would be instructive to examine its behavior.Figure 5a demonstrates the real and imaginary parts of P 0 of the used NR-Line with length L nr = 9.4 mm over frequency.It is seen from Fig. 5a that P 0 differs from unity over the entire frequency band.

Analysis
In order to examine the effect of L nr on the extracted S D 11 , S D 21 , S D 12 , and S D 22 , we also extracted these S-parameters for the constructed bianisotropic MM slab by our method using an NR-Line (an empty waveguide section) with L nr = 10.16 mm. Figure 6a-f 4a-f (with maximum variation less than 3%).This indicates not only the non-dependence of our method on L nr (provided that P 0 = 1.0 ) but also its stability.

Extracted S-parameters of a sensing area
The examined topology After validating our proposed method for a bianisotropic MM slab positioned into a waveguide section, we then proceeded with extraction of S-parameters of a sensing area (or a two-port network (line)) involving SRR resonators next to a microstrip line.Figure 7a-c illustrate photos of the fabricated configurations of a R-Line, a NR-Line, and a device with double SRR resonators next to the microstrip line (grounds are not shown for clarity).
The FR4 material with ε r,sub = 4.3(1 − i0.025) and thickness d sub = 1.6 mm was used as a substrate material.Microstrip line, R-Line, NR-Line, Device, and ground were all constructed by the copper material ( σ = 5.8 × 10 7 S/m and t m = 35µm).For the R-Line, microstrip line with a width of 10.0 mm and a length of L r = 9.7 mm ( w s = 3.0 mm, see Fig. 1) was considered.This line having an effective relative dielectric constant of approximately ε eff ∼ = 3.618 − i0.085 and an effective impedance of approximately Z eff ∼ = 21.881+ i0.258 ohm 55 introduces symmetric reflections on both sides of the microstrip line.
For the NR-Line, we considered a microstrip line with a width of w s = 3.0 mm (see Fig. 1), and a length of L nr = 9.7 mm.This line having an effective relative dielectric constant of approximately ε eff,0 ∼ = 3.263 − i0.074 and an effective impedance of approximately 50.573 + i0.571 ohm 55 produces essentially near-zero reflection.Figure 5b shows the magnitudes of simulated S-parameters of the configuration of the NR-line next to the R-Line in Fig. 7b.For microstrip measurements, the setup in Fig. 3c, except for the waveguide sections, was utilized.In the simulations, the Frequency-Domain solver of the CST Microwave Studio was utilized.Here, E t = 0 was set at the ground, open boundary conditions with additional space was used on the top, and open boundary conditions (without additional space) were applied for all configurations in Fig. 7a-c.Waveguide ports whose dimensions were calculated using the built-in macro function of port extension coefficient were positioned at beginning of the microstrip lines.Adaptive mesh refinement was activated in the solver with an accuracy of 10 −12 (3 rd order solver).It is seen from Fig. 5b that the configuration in Fig. 7b  For the device, two identical resonators (next to the microstrip line) cascaded in longitudinal direction were considered.The geometrical parameters of this device, as shown in Fig. 1, are as follows: L r1 = 9.7 mm, L r2 = 12.7 mm, w = g = 0.9 mm, s = 1.2 mm, L g1 = 0.40 mm, and L g2 = 0.50 mm.Besides, the geometrical parameters of the microstrip line section are L s1 = 15.85 mm, L s2 = 12.85 mm ( w s = 3.0 mm), and L s3 = 2.15 mm (the same for the R-Line and NR-Line configurations in Fig. 7a,b. Besides, Fig. 7d,e illustrate, respectively, the spatial distributions of electric field around the SRRs (side view) and surface current on the surface of the metals of SRRs (side view) at 2.193 GHz where |S 21 | has a minimum value.It is seen from Fig. 7d,e that aside from circulating currents, which augment specially for the SRR segment producing such a difference according to the configuration in Fig. 1.First, the SMA tapers or launchers used to transfer the coaxial line energy to the microstrip lines alter both magnitudes and phases of S-parameters.Second, microstrip feedline straights mainly influence phases of S-parameters.Third, microstrip feedline bends introduce changes chiefly in the magnitudes of S-parameters.

Extracted S-parameters of the sensing area
To eliminate the effect of the SMA connectors on measurements, and to extract only the S-parameters of the sensing area in Fig. 1, we implemented our proposed method using uncalibrated S-parameters of the configurations in Fig. 7a-c.Figure 9a-f  of the sensing area over 1.5-2.5 GHz (after applying the RA procedure to the extracted S-parameters for a frequency range of 10 MHz using N int = 10 (selected as a higher value than the one used in the study 54 to get more smoothed measurement data), f max = 2.5 GHz, f min = 1.5 GHz, and N f = 1001 54 ).Extracted S-parameters without the RA are not presented here for simplicity.For comparison, in addition to S-parameter simulations, we applied the TRL calibration procedure 22 and the de-embedding methods [46][47][48][49] .It is noted that the de-embedding method is restricted to S D 21 and S D 12 only.In implementation of the TRL calibration procedure, a calibration kit designed using an FR4 substrate ( ε r,sub = 4.3(1 − i0.025) and d sub = 1.6 mm), as shown in Fig. 3d, was utilized.For the thru standard, a 60 mm microstrip line ( w s = 3.0 mm and Z eff ∼ = 50 ) was used.For the line standard, a 73.83 mm microstrip line ( w s = 3.0 mm and Z eff ∼ = 50 ), which corresponds to an effective length of 13.83 mm in reference to the thru standard.This line standard, in reference to the thru standard, will produce an effective bandwidth of 4.44 GHz (between 560 MHz and 5.0 GHz), within which the line phase undergoes a maximum change of ∓90 o56 .The reflect line was implemented by a well-soldered via.An additional microstrip with a sufficient length of 30 mm was used for all standards to measure smoother S-parameters after the TRL calibration.
Among the applied methods 22 and [46][47][48][49] , only the results from the methods in 22 and 47 are presented in Fig. 9a-f for a clear view.A quantitative analysis of all the methods 22 and 46-49 will be presented shortly.It is noted from Fig. 9a-f that the simulated, measured, and extracted S-parameters of the sensing area are close each other over the entire band.We think that small oscillations observed in the extracted real and imaginary parts of S D 11 , S D

21
( ∼ = S D 12 ), and S D 22 of the sensing area might be partly due to tolerances in the fabricated configurations of the R-Line, the NR-Line, and the sensing area.For a quantitative analysis for how well the extracted or measured S-parameters approach the simulated ones, we calculated the root-mean-square error (RMSE) values for all considered methods using

Advantages and disadvantages of the proposed method
Table 1 presents a comparison of our method with two calibration techniques (SOLT and TRL (or LRL)) 21,22 and with other de-embedding techniques in the studies [46][47][48][49] in terms of the need for error network analysis, the total number of standards used in their implementation, capability of full two-port characterization, the possibility of any sign ambiguity, realization of standards, and requirement of a new design if a new two-port network or line is utilized.The following points are noted from the results in Table 1.First, our de-embedding technique, just as other de-embedding techniques in the studies [46][47][48][49] , does not require determination of error networks in the characterization of a two-port network (transmission line or sample), whereas calibration techniques SOLT and TRL (or LRL) (as well as other calibration techniques) do require this determination.Second, while our method and the de-embedding techniques in the studies [46][47][48][49] necessitate two different standards in their implementation, the calibration techniques SOLT and TRL (or LRL) (as well as other calibration techniques) need at least three different calibration standards for their application.Third, our de-embedding technique, the de-embedding techniques in the studies [47][48][49] , and calibration techniques SOLT and TRL (or LRL) can perform full two-port characterization.Nonetheless, the de-embedding technique 46 is limited to S 21 and S 12 only.Fourth, our de-embedding technique together with calibration techniques SOLT and TRL (or LRL) do not have any sign (41)   ambiguity in the full characterization procedure (determining all S-parameters) of a two-port network or line.On the other hand, the de-embedding techniques [46][47][48][49] could have such an ambiguity problem.Fifth, while standards of our de-embedding technique and other de-embedding techniques along with the calibration technique 22 are relatively easier to realize than those of the calibration technique 21 , because the realization of the open standard could be partly harder.It should be pointed out here that as the calibration techniques SOLT and TRL (or LRL) (as well as other calibration techniques), the accuracy of our proposed method and the de-embedding techniques 46-49 is mainly related to non-unity value of P 0 .To eliminate this disadvantage, as a rule of thumb, shorter NR-Lines, which can be arranged in the design procedure once the frequency range is specified, should be used to remove this possibility.Finally, the proposed method and the de-embedding techniques [46][47][48][49] share the common problem of the requirement of a new design if the two-port line modifies (e.g., if the feedline of the microstrip line changes).The calibration techniques SOLT and TRL (or LRL) (as well as other calibration techniques) do not have such a problem.Nonetheless, such a drawback is not the main issue in the sensing area characterization of sensors since, once designed, optimized, and then fabricated, these sensors are utilized only for a precise application [13][14][15][16][17][18][19][20] .

Conclusion
A method is proposed to determine the S-parameters of two-port devices (or networks) using uncalibrated S-parameter measurements at microwave frequencies.The method requires the use of non-reflecting line and reflecting line standards (partially unknown) and determines uniquely all S-parameters of a two-port device without the need for evaluating error coefficients or networks.The method is first validated by S-parameters of a bianisotropic MM slab (constructed by square-shaped SRRs embedded into a waveguide) as the first device.After, it is tested for extracting S D 11 , S D 12 , S D 21 , and S D 22 of a sensing area involving double SRRs next to a microstrip line.The TRL calibration procedure and four different de-embedding techniques, supported by S-parameter simulations, were applied to examine the accuracy and performance of our method.Our method, however, requires measurements of two (direct and reversed) configurations of the device.Eliminating this need will be considered for a future study.

Figure 2 .
Figure 2. Measurement configurations: (a) A reflecting line (R-Line) between error networks X and Y, (b) a non-reflecting line (NR-Line) next to the R-Line between X and Y, (c) and (d) direct and reversed connections of a device between X and Y, and (e) the reversed configuration in (b).

Figure 4 . 21 ∼ = S D 12
Figure 4. Simulated S-parameters ('Sim.' with solid lines), measured S-parameters after the TRL calibration technique ('Meas.(TRL)' with dashed lines), and extracted S-parameters by the proposed method for L nr = 9.4 mm ('Ext.(PM) Without RA' by dashdot lines for the result without RA and 'Ext.(PM) With RA' by dotted lines for the result with RA) of the constructed bianisotropic MM slab.(a) Real and (b) imaginary parts of S D 11 , (c) real and (d) imaginary parts of S D 21 ( ∼ = S D 12 ), and (e) real and (f) imaginary parts of S D 22 .
illustrate the extracted S D 11 , S D 21 ( ∼ = S D 12 ), and S D 22 after applying the RA procedure for frequency range of approximately 42 MHz.It is noted from Fig. 6a-f that the extracted S D 11 , S D 21 , and S D 22 for L nr = 10.16 mm are similar to those for L nr = 9.4 mm given in Fig.

Figure 5 .
Figure 5. (a) Real and imaginary parts of measured P 0 for the NR-Line (an empty waveguide section) with L nr = 9.4 mm and the NR-Line composed of a microstrip line with L nr = 9.7 mm and (b) magnitudes of simulated S-parameters of the configuration in Fig. 7b.
show the extracted real and imaginary parts of S D 11 , S D 21 , S D 12 , and S D 22

Figure 7 .
Figure 7. Fabricated microstrip lines: (a) The configuration of the R-Line in Fig. 2a, b the configuration of the NR-line next to the R-Line in Fig. 2b, c the configuration of the Device or the sensing area (double resonators next to the microstrip line) in Fig. 2c, and spatial distributions of (d) electric field (V/m) around the SRRs (side view) and (e) surface current (A/m) on the surface of the metals of SRRs (side view) at 2.193 GHz.